arxiv: 2605.12829 · v1 · submitted 2026-05-12 · ⚛️ physics.soc-ph · cs.SY· eess.SY

Recognition: no theorem link

Optimal excitation and measurement patterns for networks with tree topology

Alexandre Sanfelici Bazanella, Eduardo Mapurunga

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Pith reviewed 2026-05-14 19:14 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SYeess.SY

keywords treeaccuracymatrixnetworksbestempsexcitationexcite

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The pith

For networks with tree topology, minimal excitation and measurement patterns selected via the partial information matrix optimize the trace of the asymptotic covariance matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that in dynamic networks arranged as trees, one can choose the smallest number of nodes to excite and measure while achieving the best possible accuracy in estimating the module parameters. This matters because it allows efficient identification of large networks without exciting or measuring every node. The approach uses a partial information matrix to build the necessary information matrix for any pattern. For a class of cross trees, the estimation accuracy for a module turns out to depend on the size of its parameters, and exciting is preferable when parameters are similar. Numerical examples confirm these guidelines work as a practical selection tool.

Core claim

The central claim is that the partial information matrix concept enables systematic selection of minimal EMPs for tree networks that minimize the trace of the asymptotic covariance matrix. For cross trees, module accuracy depends on parameter magnitudes, with excitation preferred when magnitudes are equal. The paper extends topological conditions under which a module's accuracy is independent of other tree parameters.

What carries the argument

The partial information matrix, a tool to construct the full information matrix for any excitation and measurement pattern in a dynamic network by considering only the relevant nodes in the tree structure.

Load-bearing premise

The network must be precisely a tree with no cycles, and the asymptotic covariance matrix must give an accurate picture of the actual estimation errors for the chosen patterns.

What would settle it

Run a finite-sample simulation on a cross tree network using the proposed minimal EMP and a non-minimal one; if the non-minimal pattern yields a smaller trace of the sample covariance matrix than the asymptotic prediction for the minimal one, the optimality claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.12829 by Alexandre Sanfelici Bazanella, Eduardo Mapurunga.

**Figure 1.** Figure 1: Two classes of tree networks. The network matrix for the cross and the inverted cross are given by Ac =      0 0 0 0 0 a 0 21 0 0 0 0 0 a 0 32 0 0 0 0 a 0 42 0 0 0 0 a 0 52 0 0 0      , AT i =      0 0 0 a 0 41 0 0 0 0 a 0 42 0 0 0 0 a 0 43 0 0 0 0 0 a 0 54 0 0 0 0 0      . (10) Let us start our analysis with the cross network. Notice that node 1 is a source, while nodes 3, 4, 5 are sin… view at source ↗

**Figure 2.** Figure 2: Percentage of the winning EMPs for the trees with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

In this work we evaluate the excitation and measurement patterns (EMP) for networks with tree topology. We investigate guidelines for the selection of the minimal EMPs, i.e. those with the least number of excited and measured nodes combined, for which the accuracy obtained, in terms of the trace of the asymptotic covariance matrix, is optimal. We introduce the concept of partial information matrix as a means to systematically obtain the information matrix for any dynamic network. For a specific tree class, called cross, we show that the accuracy of a particular module depends on the magnitude of the parameters to be estimated. Furthermore, when all factors are equal, it is best to excite. %we show that for small magnitudes of this parameter, it is best to excite. We extend a topological condition for branches under which the accuracy of a particular module of the network is independent of the other parameters from the tree. We provide a numerical analysis showing that our guidelines could be used as a selection tool for minimal EMPs for tree networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives practical guidelines for minimal EMPs in tree networks via partial information matrix but relies on asymptotic optimality without finite-sample verification.

read the letter

This paper shows how to pick minimal excitation and measurement patterns for tree networks to optimize the trace of the asymptotic covariance using a partial information matrix, and finds that accuracy in cross trees depends on parameter magnitudes. They introduce the partial information matrix as a systematic way to obtain the information matrix for dynamic networks. That is a useful addition because it avoids having to rebuild the matrix from scratch for each choice of excited and measured nodes. For cross trees they observe the dependence of module accuracy on parameter magnitude, and when magnitudes are equal it is better to excite. They also extend the topological condition under which a module's accuracy is independent of other parameters. The numerical examples demonstrate the guidelines in action. The paper does well at providing a clear selection tool rather than leaving it to trial and error. The soft spot is the reliance on asymptotic results. The analysis minimizes the asymptotic trace but does not verify whether those patterns perform best in finite samples through repeated estimations. This matters for the practical guidelines they propose. The work stays within exact tree topologies. This is for researchers focused on experiment design in network identification. Readers working with tree graphs would get concrete selection rules from it. I would recommend sending it for peer review. The new tool and observations are worth a closer look even if more finite-sample checks would help.

Referee Report

2 major / 1 minor

Summary. The manuscript develops guidelines for selecting minimal excitation and measurement patterns (EMPs) in dynamic networks whose topology is a tree. Using a newly introduced partial information matrix, the authors derive the information matrix for any such network and select the minimal EMPs that minimize the trace of the asymptotic covariance matrix of the module-parameter estimates. For the subclass of cross trees they prove that the accuracy of a given module depends on the magnitude of its parameters; they also extend a topological condition under which the accuracy of one module is independent of all other parameters. A numerical study is presented to illustrate that the derived guidelines can serve as a practical selection tool.

Significance. If the asymptotic optimality criterion is shown to rank patterns reliably under finite-sample conditions, the work supplies a systematic, topology-aware method for minimizing experimental effort in network identification. The partial-information-matrix construction and the explicit dependence results for cross trees are potentially reusable in other structured identification problems.

major comments (2)

[Numerical analysis section] The central optimality claim rests on minimization of the trace of the asymptotic covariance matrix obtained from the partial information matrix. The numerical analysis evaluates this asymptotic trace directly; no Monte-Carlo comparison of the resulting EMP rankings against empirical covariance matrices obtained from finite-length data is reported. Because the manuscript itself notes that module accuracy in cross trees depends on parameter magnitudes (which affect conditioning), the absence of finite-sample verification leaves the practical utility of the selection guidelines unconfirmed.
[Section introducing the partial information matrix] The partial information matrix is introduced as the key device for obtaining the information matrix of an arbitrary tree network. The manuscript does not supply an explicit recursive or closed-form expression that would allow a reader to reconstruct the matrix entries from the tree structure and the chosen EMP; without this, the claim that the construction is systematic for any tree cannot be verified from the given material.

minor comments (1)

[Abstract] The sentence 'when all factors are equal, it is best to excite' is ambiguous; it should be clarified whether this refers to equal parameter magnitudes, equal noise variances, or equal branch lengths.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the constructive comments. We address the two major comments point by point below and indicate the revisions we will make.

read point-by-point responses

Referee: [Numerical analysis section] The central optimality claim rests on minimization of the trace of the asymptotic covariance matrix obtained from the partial information matrix. The numerical analysis evaluates this asymptotic trace directly; no Monte-Carlo comparison of the resulting EMP rankings against empirical covariance matrices obtained from finite-length data is reported. Because the manuscript itself notes that module accuracy in cross trees depends on parameter magnitudes (which affect conditioning), the absence of finite-sample verification leaves the practical utility of the selection guidelines unconfirmed.

Authors: We agree that the absence of finite-sample Monte-Carlo verification is a limitation, particularly given the noted dependence of module accuracy on parameter magnitudes in cross trees. The numerical study in the manuscript is deliberately focused on the asymptotic trace criterion, which is the standard theoretical benchmark in system identification. To strengthen the practical utility, we will add a short Monte-Carlo experiment on one representative cross-tree example in the revised numerical section, comparing the asymptotic EMP rankings against empirical covariance estimates obtained from finite-length data. revision: yes
Referee: [Section introducing the partial information matrix] The partial information matrix is introduced as the key device for obtaining the information matrix of an arbitrary tree network. The manuscript does not supply an explicit recursive or closed-form expression that would allow a reader to reconstruct the matrix entries from the tree structure and the chosen EMP; without this, the claim that the construction is systematic for any tree cannot be verified from the given material.

Authors: We thank the referee for highlighting this presentational gap. While the partial information matrix is defined and its role in obtaining the network information matrix is explained, we acknowledge that an explicit reconstruction procedure is not provided. In the revised manuscript we will insert a dedicated subsection containing a recursive algorithm that computes the partial-information-matrix entries directly from the tree topology and the chosen excitation/measurement pattern, thereby making the systematic construction fully verifiable. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via standard asymptotic analysis

full rationale

The paper introduces the partial information matrix as a computational tool to obtain the information matrix for tree networks and derives EMP selection guidelines by minimizing the trace of the resulting asymptotic covariance matrix. No quoted step reduces a claimed prediction or optimality condition to a fitted parameter by construction, nor does any load-bearing premise collapse to a self-citation whose validity is assumed rather than re-derived. The topological independence conditions and cross-tree parameter-magnitude results follow directly from the matrix structure without renaming known results or smuggling ansatzes. The chain is therefore independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard asymptotic covariance results from system identification theory and the assumption that the network graph is exactly a tree. The partial information matrix is introduced as a computational device rather than a new physical entity.

axioms (2)

standard math Asymptotic covariance of parameter estimates is given by the inverse of the information matrix derived from excitation and measurement patterns
Invoked throughout the abstract as the accuracy metric
domain assumption The network graph is exactly a tree with no cycles
Stated as the topology class under study

invented entities (1)

partial information matrix no independent evidence
purpose: To systematically obtain the information matrix for any dynamic network without constructing the full matrix
Introduced as a new conceptual tool for EMP selection

pith-pipeline@v0.9.0 · 5487 in / 1477 out tokens · 42388 ms · 2026-05-14T19:14:37.954390+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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